Iodine dynamics

I summarized the math in the last post. Here is what you can do with it.

I₂ can be described with Morse potentials. Here we are looking at the electronic ground state and the second excited state (because that is the bright state). Energy is plotted against I-I distance. I guess it is pretty much straight forward to solve the stationary Schrödinger equation of such a system to get the vibrational levels and wave functions. The lowest 20 ground state levels and 80 excited state levels are shown.
This is the textbook situation of a larger equilibrium bond distance and lower binding energy in the excited state. Vertical excitation from the ground state will put the wave packet on the left edge of the potential in a highly excited vibrational state. In the classical picture you can say the molecule will start vibrating because the two atoms are closer together than the new equilibrium distance.

Excitation from the v=0 vibronic ground state would require about 509 nm excitation energy and would lead to an almost dissociative state. In the simulation we started with v=1 where you have more density at a larger bond distance and a vertical excitation of 588 nm from the second maximum. v=1 is a realistic situation because this level is just 213/cm or 2.5 kJ/mol above the ground state, so we have 36% in this state (relative to 100% v=0). The situation works together with what we observe. 588 nm corresponds to yellow light. Yellow absorption gives a purple appearance.

In the simulation there was a 300 fs Gauss pulse (simulated with numerical Runge-Kutta integration) and then the wave packet evolved like I explained in the last post. With this ultrashort pulse, excitation of a coherent wave packet is possible. This wave packet is formed above the second maximum of the ground state function, then it oscillates back and forth. These oscillations occur with a period of 333 fs on average, this fits very well with the energy gap between the two levels where the wave packet should be mostly localized according to the excitation energy.



The classical explanation for the oscillations is that since the pulse is only about the length of a vibrational period or shorter, you can excite coherent motion.

You get the same with quantum mechanics but with a different explanation. First you look at the energy uncertainty of the pulse:

Another formula on Wikipedia looks almost the same, the energy of a harmonic oscillator:
In other words: The energy uncertainty of a pulse corresponds to the energy gap between two levels of a system with an oscillatory period equal to the pulse length. If the pulse is much longer than the oscillatory period, excitation will be sharp - one eigenstate will be excited and all subsequent motion is only a phase factor in the complex plane. If it is the same length or shorter there will be excitation into several levels and temporal evolution like I explained last time.

The software used is: Fortran scripts for the numerical integration, the Python Pylab package for creation of graphics, Video Mach for making a movie out of the pictures.

TDSE

In a way it is easier to solve the time-dependent Schrödinger equation than the time-independent one. If you don't look too closely at H, it is a first order ordinary differential equation. And not even an eigenvalue problem.

If H is constant over time the solution is:

You are probably used to this if you have multiplication with a constant. It works the same way with a linear operator like it is the case here.

This way you can propagate a wave packet when you know the potential energy and the wavefunction at the start.

For every Hermitian operator (like H) there is a basis of its eigenfunctions. The physical interpretation is that the state of the system can always be seen as superposition of eigenstates.
Expand Ψ into these functions. And apply the fact that they are eigenfunctions.

You have time-independent functions with constant weights, only their phases change. The wave packet is the interference pattern.

The harmonic oscillator is an interesting example. The energies are E_k = (k+1/2)hν. If you plug this in, you will notice that it is a periodic function and the frequency is ν the classical frequency of the oscillator and the frequency of the light that causes the transition. (To be precise: Ψ*Ψ is the same after 2π/ν and Ψ has the opposite sign, it has a period of 4π/ν.) I will show in the next post what such a wave packet looks like.

One more consideration. What happens to an eigenfunction in the time-dependent formulation?

It shows a phase change but the physically relevant quantity Ψ*Ψ stays the same. We have a stationary solution.

Are all enantiomers equal?

The typical statement is that the physical properties of two enantiomers are exactly the same. Differences are only in interactions with other chiral substances and polarized radiation. This is based on the intuitive assumption that a mirror image of our world would behave exactly the same as our world. In physics this is called parity symmetry [1]. But interestingly parity symmetry is not always conserved.

Out of the four forces of nature, it's the weak one that steps out of the line and breaks P- (and even CP-) symmetry. As the name suggests it is weak. And it is also short range, on the order of attometers. That's why we don't notice it. Nonetheless theory predicts energy level splittings between enantiomers. This could be on the order of 10^-15 cm^-1. This is apparently extremely small and even the newest experiments haven't gotten below 10^-13 cm^-1.

Don't bet too much money on the emergence of weak force intermediated enantiospecific synthesis yet. But even though P-violation is out of every day life it seems that with improved experiments or in different systems the effect could be observed. And then chemistry could be an interesting alternative to clashing things together with higher and higher energy, at least a complimentary tool.

Another question has been experimentally adressed. What is the minimum number of atoms you need for a chiral system? It is one. Isolated Bi atoms have been shown to rotate the plane of polarised light.

I am sorry I am not quoting literature. The information comes from a lecture by Robert Berger, Stephen Hawking's "Brief History of Time", and Wikipedia.

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[1] Two other fundamental symmetries are charge and time. Conversely to separate parts, it is generally assumed that CPT symmetry is conserved in the world, i.e. that things would be the same if you simultaneously changed charge, parity and the direction of time.

[2] Yes, I am still only treating physics at the popular science level. To be honest, it was even the Illustrated Brief History of Time ...